SCIENTIA SINICA Informationis, Volume 50 , Issue 12 : 1817(2020) https://doi.org/10.1360/N112018-00272

Concept reduction and concept characteristics in formal concept analysis

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  • ReceivedOct 14, 2018
  • AcceptedAug 9, 2019
  • PublishedNov 18, 2020


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  • Figure 1

    The concept lattice corresponding to example 1

  • Figure 2

    A concept reduct $\mathcal{F}_{1}$ of example 1

  • Table 1   The context of living beings and water $(G,~M,~I)$
    $G$ $a$ $b$ $c$ $d$ $e$ $f$ $g$ $h$ $i$
    $1$$1$$1$ $0$$0$$0$$0$$1$$0$$0$
    $2$$1$$1$ $0$$0$$0$$0$$1$$1$$0$
    $3$$1$$1$ $1$$0$$0$$0$$1$$1$$0$
    $4$$1$$0$ $1$$0$$0$$0$$1$$1$$1$
    $5$$1$$1$ $0$$1$$0$$1$$0$$0$$0$
    $6$$1$$1$ $1$$1$$0$$1$$0$$0$$0$
    $7$$1$$0$ $1$$1$$1$$0$$0$$0$$0$
    $8$$1$$0$ $1$$1$$0$$1$$0$$0$$0$
  • Table 2   The concept lattice corresponding to the formal context in example 1
    Label Concept Label Concept Label ConceptLabel ConceptLabel Concept
    $c_{1}$$(\emptyset,M)$ $c_{5}$$(6,~abcdf)$$c_{9}$$(36,~abc)$$c_{13}$$(234,~agh)$$c_{17}$$(12356,~ab)$
    $c_{2}$$(4,~acghi)$ $c_{6}$$(34,~acgh)$$c_{10}$$(678,~acd)$$c_{14}$$(568,~adf)$$c_{18}$$(5678,~ad)$
    $c_{3}$$(3,~abcgh)$ $c_{7}$$(23,~abgh)$$c_{11}$$(68,~acdf)$$c_{15}$$(1234,~ag)$$c_{19}$$(G,~a)$
    $c_{4}$$(7,~acde)$ $c_{8}$$(123,~abg)$$c_{12}$$(56,~abdf)$$c_{16}$$(34678,~ac)$
  • Table 3   The classification of example 1
    Type Concept
    Core concept $c_{2}$, $c_{4}$
    Relatively necessary concept$c_{3}$, $c_{6}$, $c_{7}$, $c_{8}$, $c_{9}$, $c_{10}$, $c_{11}$, $c_{12}$, $c_{13}$, $c_{14}$, $c_{15}$, $c_{16}$, $c_{17}$
    Unnecessary concept $c_{1}$, $c_{5}$, $c_{18}$, $c_{19}$
  • Table 4   A part of pairs and concepts
    Serial number $k$Pair Concept set ${\rm~CS}_{k}$Serial number $k$Pair Concept set ${\rm~CS}_{k}$
    1$(1,~b)$ $\{c_{8},~c_{17}\}$8$(3,~g)$ $\{c_{3},~c_{6},~c_{7},~c_{8},~c_{13},~c_{15}\}$
    2$(1,~g)$ $\{c_{8},~c_{15}\}$9$(3,~h)$ $\{c_{3},~c_{6},~c_{7},~c_{13}\}$
    3$(2,~b)$ $\{c_{7},~c_{8},~c_{17}\}$10$(5,~b)$$\{c_{12},~c_{17}\}$
    4$(2,~g)$ $\{c_{7},~c_{8},~c_{13},~c_{15}\}$11$(5,~f)$$\{c_{12},~c_{14}\}$
    5$(2,~h)$ $\{c_{7},~c_{13}\}$12$(8,~c)$$\{c_{10},~c_{11},~c_{16}\}$
    6$(3,~b)$ $\{c_{3},~c_{7},~c_{8},~c_{9},~c_{17}\}$13$(8,~f)$$\{c_{11},~c_{14}\}$
    7$(3,~c)$ $\{c_{3},~c_{6},~c_{9},~c_{16}\}$
  • Table 5   Pairs and concepts corresponding to the minimum concept intervals
    Serial number $k$ Minimum concept interval Pair Concept set ${\rm~CS}_{k}$
    1$\mathcal{I}_{1b}$$(1,~b)$ $\{c_{8},~c_{17}\}$
    2$\mathcal{I}_{1g}$$(1,~g)$ $\{c_{8},~c_{15}\}$
    5$\mathcal{I}_{2h}$$(2,~h)$ $\{c_{7},~c_{13}\}$
    7$\mathcal{I}_{3c}$$(3,~c)$ $\{c_{3},~c_{6},~c_{9},~c_{16}\}$